State and Find a relation for potential difference

Potential Difference

State and find a relation for potential difference.

Potential difference: Potential difference between any two points in an electric field is defined as the work done in moving a unit positive charge from one point to the other point against the electric force of the electric field irrespective of the path followed.

Potential difference between any two points in an electric field is defined as the electric potential energy difference per unit charge.

Let $\Delta U = U_B - U_A$ be the electrostatic potential energy difference between two points A and B in the electric field. The electric potential difference $\Delta V$ between points A and B is given by

$\Delta V = V_B - V_A = \frac{\Delta U}{q_0}$

$\Delta V = \frac{U_B - U_A}{q_0} = \frac{W_{AB}}{q_0}$

Now,

$U_B - U_A = -q_0 \int_A^B \vec{E} \cdot d\vec{r}$

$\Delta U = -q_0 \int_A^B \vec{E} \cdot d\vec{r}$

\Delta V = V_B - V_A = \frac{\Delta U}{q_0}$

$= \frac{-q_0 \int_A^B \vec{E} \cdot d\vec{r}}{q_0} = -\int_A^B \vec{E} \cdot d\vec{r}$

SI Unit of potential difference is volt (V) and $1 V = 1 J C^{-1}$